Chapter 7.2, Problem 57E

a.

To determine

To find:The least squares regression line $y=ax+b$ for the data.

Answer to Problem 57E

The least squares regression line $y=ax+b$ for the data is $y=14x+19$ .

Explanation of Solution

Given information:

An agricultural scientist used four test plots to determine the relationship between wheat yield $y$ (in bushels per acre) and the amount of fertilizer $x$ (in hundreds of pounds per acre). The table shows the results

  

System of equations to find the values of $a$ and $b$ is:

  $\{\begin{array}{c}4b+7a=174\\ 7b+13.5a=322\end{array}$

Concept Used:

To find the least squares regression line $y=ax+b$ for a set of points $(x_1,y_1),(x_2,y_2),\mathrm{....},(x_n,y_n)$ is by solving the system below for $a$ and $b.$

  $\{\begin{array}{c}ab+\left({\displaystyle \sum _{i=1}^n{x}_i}\right)a=\left({\displaystyle \sum _{i=1}^n{y}_i}\right)\\ \left({\displaystyle \sum _{i=1}^n{x}_i}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)a=\left({\displaystyle \sum _{i=1}^n{x}_i{y}_i}\right)\end{array}$

Calculations:

Solving the system of equations using substitution method, we have:

  $\begin{array}{l}4b+7a=174\\ \Rightarrow 4b=174-7a\\ \Rightarrow b=\frac{174-7a}{4}\end{array}$

Substituting the above finding in the second equation, we get

  $\begin{array}{l}7\left(\frac{174-7a}{4}\right)+13.5a=322\\ \Rightarrow 304.5-12.25a+13.5a=322\\ \Rightarrow 1.25a=17.5\\ \Rightarrow a=\frac{17.5}{1.25}\\ \Rightarrow a=14\end{array}$

Now, using the above finding value:

  $\begin{array}{l}b=\frac{174-7a}{4}\\ \Rightarrow b=\frac{174-7(14)}{4}\\ \Rightarrow b=\frac{76}{4}\\ \Rightarrow b=19\end{array}$

Conclusion:

Using the graphing utility, we obtain the following results:

  

Therefore, the results we got are correct.

b.

To determine

To find: Theestimatedvalue of yield for a fertilizer application of 160 pounds per acre.

Answer to Problem 57E

The wheat yield for a fertilizer application of 160 pounds per acre is of 41 bushels per acre.

Explanation of Solution

Given information:

An agricultural scientist used four test plots to determine the relationship between wheat yield $y$ (in bushels per acre) and the amount of fertilizer $x$ (in hundreds of pounds per acre). The table shows the results

  

System of equations to find the values of $a$ and $b$ is:

  $\{\begin{array}{c}4b+7a=174\\ 7b+13.5a=322\end{array}$

Calculations:

From the previous part, we got the linear model $y=14x+19$ .

At 160 pounds of fertilizer, the value of $x=1.60$ .

Thus,

  $\begin{array}{l}y=14(1.60)+19\\ \Rightarrow y=22.4+19\\ \Rightarrow y=41.4\end{array}$

Therefore, the wheat yield for a fertilizer application of 160 pounds per acre is of 41 bushels per acre.